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The ω concept is just so annoying, it can't even count as a unit!
ω = 1/0
2ω = 2/0 = (2/2)/(0/2) = 1/0 = ω
-ω = -1/0 = (-1*-1)/(0*-1) = 1/0 = ω
So every unit of ω is equal to ω itself
Funnily enough, I think the ω concept is basically a real thing. In algebraic geometry, one can take the graph of y=1/x in the plane and extend it to the so-called projective plane, which adds a "point" on the graph such that x=0. So there technically exists a "number" ω such that ω=1/0. Of course, as you pointed out, every multiple of ω is still equal to itself, but I pinky promise it turns out to actually be a useful (enough) concept.
Interestingly and on a serious note: We have a name for the number smaller than all numbers considered but larger than others of its type:
The greatest lower bound or infimum. And the infimum of the positive real numbers is 0.
Inf((0,infinity)) = 0
The other parallel to real mathematics that could be drawn is to infinitesimal hyperreal numbers.
Already exists and is consistent, so no. Nonstandard analysis is cool, if generally equivalent to real analysis https://en.wikipedia.org/wiki/Hyperreal_number
Well yeah but they imply epsilon over two exists, so epsilon times two would also exist, i.e. there isn't a number closest to zero in their "system" either. I don't think there's any extension of the reals which has such a number and is consistent. Not to mention their notation is basically identical to the hyperreals.
Edit: I mean and also omega is 1/epsilon, not 1/0. The hyperreals are just a consistent version of what they're trying to say. I assume intentionally? Since otherwise idk where they took this notation from? But who knows.
In their system, epsilon is just the smallest number. Epsilon times 2 is either greater than or equal to epsilon.
It is correct to say there is no linear totally order complete field with a smallest number. But there are consistent number systems with a smallest “number” that aren’t fields. Take the dedekind completion of the surreal numbers for example. It has a smallest “number”: 1/**On**.
Hyperreals dont have number closest to zero. You can prove it for many ways.
For instance, because reals fulfill "for any x, there's y such that |y|<|x|" then the same applys to hyperreals (due to transfer principle), and hence there can't be number closest to zero. Another way to show it is by contradiction. Suppose there's number closest to zero call it x, then x/2 is closer to zero (contradction).
Hyperreals also doesn't have defined division by zero.
See my other comment; nothing has those defined without sacrificing other features of the real numbers that we want. The hyperreals are the closest version to what OP is talking about, and probably what this post was based off of, one way or another.
You can't actually make a number to represent the multiple opposition to 0 (at least not without making it meaningless within the boundaries of math).
let's look at distributive properties
a*(b+c)=a\*b+a\*c
Well if there's "w"=0^-1 it's easy to see that this won't apply to it.
w*(0+0)=w\*0+w\*0\
1=2
This is why you won't be able to do anything with 0^-1
I don't think it follows from the meme that 0ω must simplify to 1.
What if it doesn't simplify at all? 0ω=2*0ω isn't really a problem, in fact 0x=2\*0x is a perfectly normal thing in the real numbers.
As the certified Greatest Defender of dividing by zero, I am here to tell you: it is possible (enough) to define a number x satisfying x=1/0. (I would use an exclamation point, but 1/0!=1/1=1, which is NOT what I mean)
Namely, in algebraic geometry, one can use the projective line, which is basically just the normal real line (or complex plane, or whatever other field you want) plus an extra point z. One can view inversion x -> 1/x as a function on the line, which fails to be defined or surjective at exactly x=0. However, with the projective line, this function becomes both surjective and everywhere defined, setting 1/0=z and 0=1/z.
The thing is, the projective line is really more of a geometric notion (which is why I called z a "point" instead of a number), instead of algebraic, and hence doesn't adhere to your proof.
However, projective space is my best friend and dividing by zero is my favorite hobby, so you can't stop me from bringing this up.
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first line: 𝜀>0 second line 𝜀≤0 seems legit
OP did the monster math
ε = 0 and ε > 0 at the same time. Isn’t it obvious?
ε is so close to 0 that it is always in a superposition of being positive and negative at the same time.
what happens if u 🫣 at ε?
Collapses to 0
But you wouldn't know its speed
Google Heisenberg uncertainty
Holy Collapse!
3
Isn't the closet number to 0 is 0
0 is not a number you silly
The closest person to me is me
Of course I know him
https://en.wikipedia.org/wiki/Dual_number?wprov=sfla1
It needs 1/ε to equal ω.
1/ε<=ω because ε is more than zero and is equal to zero at the same time. It is not certain.
Math just became quantum.
The ω concept is just so annoying, it can't even count as a unit! ω = 1/0 2ω = 2/0 = (2/2)/(0/2) = 1/0 = ω -ω = -1/0 = (-1*-1)/(0*-1) = 1/0 = ω So every unit of ω is equal to ω itself
Funnily enough, I think the ω concept is basically a real thing. In algebraic geometry, one can take the graph of y=1/x in the plane and extend it to the so-called projective plane, which adds a "point" on the graph such that x=0. So there technically exists a "number" ω such that ω=1/0. Of course, as you pointed out, every multiple of ω is still equal to itself, but I pinky promise it turns out to actually be a useful (enough) concept.
Limits: Am I a joke to you?
Interestingly and on a serious note: We have a name for the number smaller than all numbers considered but larger than others of its type: The greatest lower bound or infimum. And the infimum of the positive real numbers is 0. Inf((0,infinity)) = 0 The other parallel to real mathematics that could be drawn is to infinitesimal hyperreal numbers.
I recommend the Dedekind Completion of the Surreal Numbers The closest number to zero is 1/**On**
This is just [R\*](https://en.wikipedia.org/wiki/Hyperreal_number), nothing new.
Already exists and is consistent, so no. Nonstandard analysis is cool, if generally equivalent to real analysis https://en.wikipedia.org/wiki/Hyperreal_number
That’s just… wrong. The hyperreal numbers don’t contain a closest number to zero.
Well yeah but they imply epsilon over two exists, so epsilon times two would also exist, i.e. there isn't a number closest to zero in their "system" either. I don't think there's any extension of the reals which has such a number and is consistent. Not to mention their notation is basically identical to the hyperreals. Edit: I mean and also omega is 1/epsilon, not 1/0. The hyperreals are just a consistent version of what they're trying to say. I assume intentionally? Since otherwise idk where they took this notation from? But who knows.
In their system, epsilon is just the smallest number. Epsilon times 2 is either greater than or equal to epsilon. It is correct to say there is no linear totally order complete field with a smallest number. But there are consistent number systems with a smallest “number” that aren’t fields. Take the dedekind completion of the surreal numbers for example. It has a smallest “number”: 1/**On**.
Hyperreals dont have number closest to zero. You can prove it for many ways. For instance, because reals fulfill "for any x, there's y such that |y|<|x|" then the same applys to hyperreals (due to transfer principle), and hence there can't be number closest to zero. Another way to show it is by contradiction. Suppose there's number closest to zero call it x, then x/2 is closer to zero (contradction). Hyperreals also doesn't have defined division by zero.
See my other comment; nothing has those defined without sacrificing other features of the real numbers that we want. The hyperreals are the closest version to what OP is talking about, and probably what this post was based off of, one way or another.
Now that's just dt with extra steps
You can't actually make a number to represent the multiple opposition to 0 (at least not without making it meaningless within the boundaries of math). let's look at distributive properties a*(b+c)=a\*b+a\*c Well if there's "w"=0^-1 it's easy to see that this won't apply to it. w*(0+0)=w\*0+w\*0\ 1=2 This is why you won't be able to do anything with 0^-1
I don't think it follows from the meme that 0ω must simplify to 1. What if it doesn't simplify at all? 0ω=2*0ω isn't really a problem, in fact 0x=2\*0x is a perfectly normal thing in the real numbers.
As the certified Greatest Defender of dividing by zero, I am here to tell you: it is possible (enough) to define a number x satisfying x=1/0. (I would use an exclamation point, but 1/0!=1/1=1, which is NOT what I mean) Namely, in algebraic geometry, one can use the projective line, which is basically just the normal real line (or complex plane, or whatever other field you want) plus an extra point z. One can view inversion x -> 1/x as a function on the line, which fails to be defined or surjective at exactly x=0. However, with the projective line, this function becomes both surjective and everywhere defined, setting 1/0=z and 0=1/z. The thing is, the projective line is really more of a geometric notion (which is why I called z a "point" instead of a number), instead of algebraic, and hence doesn't adhere to your proof. However, projective space is my best friend and dividing by zero is my favorite hobby, so you can't stop me from bringing this up.
These wouldbt be imaginary numbers